Question
Write an equation of the ellipse with foci at (0, ±12), and vertices at (0, ±13). (1 point) Responses Image with alt text: Choice 1 Image with alt text: Choice 2 Image with alt text: Choice 3
Answers
GPT 3.5
The equation of an ellipse with foci at (0, ±12) and vertices at (0, ±13) can be written in the form:
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)
where \(2a = 26 \rightarrow a = 13\) and \(2c = 24 \rightarrow c = 12\), so \(b = \sqrt{a^2-c^2} = 5\)
Therefore, the equation of the ellipse is:
\(\dfrac{x^2}{169} + \dfrac{y^2}{25} = 1\)
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)
where \(2a = 26 \rightarrow a = 13\) and \(2c = 24 \rightarrow c = 12\), so \(b = \sqrt{a^2-c^2} = 5\)
Therefore, the equation of the ellipse is:
\(\dfrac{x^2}{169} + \dfrac{y^2}{25} = 1\)